Index
Adaptive coupled-oscillator array beamforming, 317–320
Adler’s formalism
coupled oscillator arrays, alternative perturbation models, external injection locking, 249–250
coupling delay, continuum models, causality violation, 149–159
Admittance matrix
multiport passive coupling networks, 55–66
single capacitor coupling network, 62–66
Affine sets, beamforming, coupled oscillator arrays, convex optimization, 297–301
All-to-all coupling phenomenon, coupling network parameters, 58–66
Amplitude perturbations, beamforming, coupled oscillator arrays, 302–307, 311–317
multi-beam pattern generation, 314–315
Antisymmetric detuning
linear arrays, continuum modeling, beam steering, 93–95
planar arrays, continuum models, Cartesian coupling, without external injection, 107–109
Aperture phase distributions
coupling delay, beam-steering, continuum model, 165–172
planar-array experimentation, 188–201
planar arrays, continuum models
Cartesian coupling, external injection locking, 109, 111–121
hexagonal coupling schemes, 123–125, 129–130, 133–137
triangular coupling schemes, 123–127
Array factor, beamforming, coupled oscillator arrays, 302–307
Array response, coupled oscillators, discrete model solution, 37
Asymmetric detuning, linear arrays, continuum model, beam steering via, 93–95
Autonomous continuous time dynamic systems, perturbation models, 220–222
Averaging theory, perturbation models
coupled oscillator systems, 231–235
Beamforming, coupled-oscillator arrays, 297–320
multi-beam pattern generation, 311–315
stability optimization, steady-state solution, 308–311
Beam-steering, coupled oscillator arrays, external injection locking, 48–50
coupling delay, continuum model, 159–172
linear array experimentation, 178–188
linear arrays, continuum model
external injection locking, 81–92
planar-array experimentation, 190–201
planar arrays, continuum models
Cartesian coupling, external injection locking, 111–120
Cartesian coupling, without external injection, 106–111
triangular and hexagonal coupling schemes, 123–137
receive-array experimentation, 201–210
unlocked state experiments, 213–215
Bessel function, coupling delay, branch point locations, 158–159
Bifurcations, nonlinear dynamical systems, 226–229
Boundary conditions
coupled oscillators, discrete model solution, 33–37
coupling delay, beam-steering, continuum model, 161–172
linear arrays, continuum modeling
beam steering via end injection, 96–102
external injection locking, 90–92
without external injection, 70–81
planar arrays, continuum models, triangular and hexagonal coupling schemes, 124–137
Branch point locations, coupling delay, continuum models, causality violation, 149–159
Bromwich contour, coupling delay
beam-steering, continuum model, 165–172
continuum models, causality violation, 149–159
Cartesian coupling, planar arrays, continuum models
with external injection, 109–118
without external injection, 103–109
Causality, and coupling delay, 139–173
beam steering and continuum model, 159–172
Cavity-backed antennas, linear array experimentation, 184–188
Central pattern generator (CPG), early systems, 3–5
Chaotic solutions, perturbation models, dynamical systems, 221–222
Chebyshev polynomials
coupled oscillators
discrete model solution, 33–37
external injection locking, 48–50
steady-state phase distribution stability, 44–46
coupling delay, discrete model, 143–147
Circuit component values, voltage-controlled oscillators, numerical simulation, 273–278
Circular polarization, phase noise experimentation, 212–213
Coaxial line stretchers, planar-array experimentation, 190–201
Complex coupling coefficient, coupling network parameters, 57–66
Conjugate phasing, receive-array experimentation, 207–210
Constant phase modulation (CPM), coupled oscillator arrays, perturbation models, 256–257
Continuation simulations, coupled oscillator arrays, 269
Continuum models
beam-steering via end detuning, 93–95
beam-steering via end injection, 95–102
external injection locking, 81–92
planar oscillator arrays, 103–137
Cartesian coupling with external injection, 109–118
Cartesian coupling without external injection, 103–109
non-Cartesian coupling topologies, 118, 121–137
Conversion matrix, coupled oscillator arrays, numerical simulation, 267–268
Convex optimization, beamforming, coupled oscillator arrays, 297–301
multi-beam pattern generation, 311–315
Correction factor, planar arrays, continuum models, triangular and hexagonal coupling schemes, 131–137
Coupled oscillator arrays (COAs)
multi-beam pattern generation, 311–315
stability optimization, steady-state solution, 308–311
current and future research issues, 321–323
discrete model solution (linearization and Laplace transformation), 31–37
external injection locking, 46–50
linear array experimentation, 177–188
mutual injection locking, 21–26
continuation methods, 269
envelope transient simulation, 268–269
externally injection-locked five-element linear coupled oscillator array, 286–288
five-element linear coupled oscillator array, 278–285
harmonic balance simulation, 266–267
periodic steady-state solutions, 270–272
harmonic radiation, extended scanning range, 288–291
periodic steady-state solutions, 270–272
self-oscillating mixer, 291–296
voltage-controlled oscillator, 272–278
overview and history of, 27–28
perturbation models
externally injection-locked systems, 247–250
phase noise experimentation, 210–213
phase solution stability, full nonlinear formulation, 41–46
planar array generalization, 50–54
receive-array experimentation, 201–210
unlocked state experimentation, 213–215
Coupling aperture size, substrate-integrated waveguide technology, 187–189
Coupling delay, causality and, 139–173
beam steering and continuum model, 159–172
Coupling networks
Cramer’s rule approach, coupling delay, discrete model, 143–147
Decay-rate optimization, coupled oscillator arrays, stability analysis, 309–311
Detuning
coupled oscillators, steady-state phase distribution solution, 38–41
coupling delay
linear array experimentation, 184–188
linear arrays without external injection, continuum model, 69–81
planar arrays, continuum models
Cartesian coupling, without external injection, 106–109
triangular and hexagonal coupling schemes, 124–137
self-oscillating mixers, receive-array experimentation, 207–210
Diagonal elements, coupled oscillators, steady-state phase distribution stability, 42–46
Difference patterns
beamforming, coupled oscillator arrays, multi-beam pattern generation, 312–315
coupled oscillators, external injection locking, 50
Differential equations
coupling delay, beam-steering, continuum model, 160–172
linear arrays, continuum model
external injection locking, 82–92
without external injection, 68–81
phase noise experimentation, perturbation models, 251–256
planar arrays, continuum models, triangular and hexagonal coupling schemes, 123–137
van der Pol differential equation, averaging method and multiple time scales, 230–231
Diffusion equations, linear arrays without external injection, 68–81
Dirac delta functions, planar arrays, continuum models, Cartesian coupling, external injection locking, 115–118
Directional couplers, harmonic radiation, numerical simulation, coupled oscillators, 288–291
Discontinuity determination, coupling delay, continuum models, causality violation, 153–159
Discrete model solution
linear arrays without external injection, 68–81
Distributed oscillators, averaging theory, coupled oscillators, 232–235
Duroid™ circuit board, planar-array experimentation, 191–201
Dynamic frequency behavior, infinite linear array, 68–70
Dynamic phase behavior, infinite linear array, 68–70
Eigenvectors/eigenvalues
beamforming, coupled oscillator arrays, 299–308
coupled oscillators
discrete model solution, 32–37
external injection locking, 47–50
steady-state phase distribution stability, 44–46
coupling delay, discrete model, 143–147
coupling network parameters, 55–66
linear arrays, continuum modeling
beam steering via end injection, 95–102
external injection locking, 90–92
without external injection, 75–81
perturbation models
equilibrium point stability, 223–226
matrix equations, steady state and stability analysis, 242–246
planar arrays, continuum models
Cartesian coupling, external injection locking, 112–118
Cartesian coupling, without external injection, 104–109
triangular and hexagonal coupling schemes, 125–137
planar oscillator arrays, 52–54
Electronic circuits, early work on, 3–5
Elliptic integrals, single capacitor coupling network, 63–66
End detuning, linear arrays, continuum model, beam steering via, 93–95
End injection, linear arrays, continuum model, beam steering via, 95–102
Ensemble frequency
coupled oscillators
discrete model solution, 36–37
steady-state phase distribution solution, 37–41
coupling delay, transmission line theory and, 139–141
linear arrays without external injection, 72–81
two oscillators, mutual injection locking, 23–26
Envelope transient simulation
coupled oscillator arrays, 268–269
periodic steady-state solution, 272
voltage-controlled oscillators, numerical simulation, 276–278
E-plane transient response, linear array experimentation, 184–188
Equilateral triangular lattice, planar arrays, continuum models, 118, 122–129
Equilibrium points
nonlinear dynamical systems bifurcations, 226–228
perturbation models
Euclidean norm, beamforming, coupled oscillator arrays, convex optimization, 299–301
Euler algorithm, coupled oscillator arrays, transient numerical simulation, 265–266
Extended scanning range, harmonic radiation, numerical simulation, coupled oscillators, 288–291
External injection locking
numerical analysis, five-element array, 286–288
linear array experimentation, 178–188
linear arrays
beam steering via end injection, 95–102
phase noise experimentation, 212–213
planar arrays, continuum models, Cartesian coupling and, 109–118
planar oscillator arrays, 54
self-oscillating mixers, numerical simulation, 294–296
Far-zone radiated fields
coupling delay, beam-steering, continuum model, 166–172
linear arrays, continuum modeling
beam steering via end detuning, 94–95
beam steering via end injection, 101–102
planar array experimentation, 190–201
planar arrays, continuum models, Cartesian coupling, without external injection, 108–109
Field effect transistor (FET) oscillators, linear array experimentation, 179–188, 212
Final value theorem, coupling delay, beam-steering, continuum model, 166–172
Finite arrays
coupling delay, beam-steering, continuum model, 159–172
planar arrays, continuum models, triangular and hexagonal coupling schemes, 118, 121–137
First-order linear differential equations, coupled oscillators, discrete model solution, 32–37
Five-element arrays
linear array experimentation, 179–180
externally injection-locked array, 286–288
Flip bifurcation, nonlinear dynamical system, periodic orbits, 229
Floquet theory, periodic steady state, nonlinear dynamical systems, 224–225
Fold bifurcation, nonlinear dynamical system, periodic orbits, 228–229
Four-element oscillator array, 200–202, 210–211
phase noise experimentation, 212–213
Fourier transforms
harmonic balance simulation, coupled oscillator arrays, 266–267
perturbation models, phase noise, 250–256
unlocked state experiments, 213–215
Free running frequencies
harmonic-balance simulation, 271–272
linear array experimentation, 178–188
perturbation models, phase noise, 250–256
planar-array experimentation, 196–201
unlocked state experiments, 213–215
van der Pol oscillator model, perturbation parameters, 236–237
voltage-controlled oscillators, numerical simulation, 274–278
Frequency-domain methods, harmonic balance simulation, coupled oscillator arrays, 266–267
Frequency doublers, harmonic radiation, numerical simulation, coupled oscillators, 288–291
Frequency triplers, planar-array experimentation, 190–201
Fundamental harmonic approximation, coupled oscillators, averaging theory, 232–235
van der Pol oscillator model, 5–7
Gaussian minimum shift keying (GMSK), coupled oscillator arrays, perturbation models, 256–257
Generator probes, self-oscillating mixers, numerical simulation, 293–296
Georgia Tech Research Institute (GTRI), planar-array experimentation, 195–201
Globally injected arrays, coupled oscillator arrays, alternative perturbation models, external injection locking, 247–250
Green’s function
coupling delay
beam-steering, continuum model, 160–172
linear arrays, continuum modeling
external injection locking, 84–92
without external injection, phase dynamics calculations, 72–81
planar arrays, continuum models
Cartesian coupling, external injection locking, 115–118
Cartesian coupling, without external injection, 105–109
triangular and hexagonal coupling schemes, 125–137
Gunn-diode oscillators, linear array experimentation, 178
Harmonic balance simulation
coupled oscillator arrays, 266–267
continuation simulations, 269
periodic steady-state solutions, 270–272
Harmonic radiation, extended scanning range, numerical simulation, coupled oscillators, 288–291
Hexagonal coupling topologies, planar arrays, continuum models, 118, 121–137
Hopf bifurcation, nonlinear dynamical systems, equilibrium points, 227–228
H-plane transient response, linear array experimentation, 184–188
Hybrid frequencies, receive-array experimentation, 207–210
Hyperplane/hyperspace geometry, beamforming, coupled oscillator arrays, convex optimization, 299–301
Impedance matrix
single capacitor coupling network, 62–66
Inequality constraints, beamforming, coupled oscillator arrays, 303–307
Injected end elements, linear arrays, continuum modeling, beam steering via, 95–102
Injection locking phenomenon
Adler’s formalism and locked/unlocked spectra, 7–21
coupled oscillator arrays, alternative perturbation models, 239–242
coupled oscillators, external injection locking, 46–50
linear array experimentation, 180–188
phase noise experimentation, 210–213
planar oscillator arrays, 50–54
self-oscillating mixers, numerical simulation, 294–296
Input signal, coupling delay, 140–141
Instantaneous frequency, injection locking phenomenon, 8–21
Integer index, oscillator coupling, 28–31
Intermediate frequency signals
receive-array experimentation, 203–210
self-oscillating mixers, numerical simulation, 292–296
Inter-oscillator phase difference
coupled oscillators
external injection locking, 46–50
steady-state phase distribution stability, 42–46
coupling delay
beam-steering, discrete model, 171–172
continuum models, causality violation, 149–159
coupling network parameters, 56–66
linear array experimentation, 179–188
linear arrays, continuum modeling
beam steering via end detuning, 93–95
beam steering via end injection, 99–102
without external injection, 77–81
Inverse Laplace transform
coupling delay
beam-steering, continuum model, 162–172
linear arrays, continuum model, external injection locking, 82–102
linear arrays without external injection, 72–81
planar arrays, continuum models, Cartesian coupling, external injection locking, 115–118
Inverse locking ranges, coupling delay
beam-steering, continuum model, 162–172
continuum models, causality violation, 150–159
Kirchhoff’s current law
coupled oscillator arrays
alternative perturbation models, 238–242
external injection locking, 247–250
harmonic balance simulation, 266–267
transient numerical simulation, 264–266
injection locking phenomenon, 8–21
van der Pol oscillator model, 5–7
perturbation parameters, 235–257
Kott sidelobe reduction system, receive-array experimentation, 203–206
Kronecker delta function
coupled oscillators, external injection locking, 46–50
Kuramoto model, coupled oscillators, averaging theory, 234–235
Kurokawa transformation, coupled oscillator arrays, alternative perturbation models, 239–242
LabView™-based display system, planar-array experimentation, 195–201
Ladder oscillators, averaging theory, coupled oscillators, 232–235
Lambert W function, coupling delay, beam-steering, continuum model, 162–172
Laplace transformation
coupled oscillators
discrete model solution, 31–37
external injection locking, 47–50
coupling delay
beam-steering, continuum model, 161–172
transmission line theory and, 139–141
linear arrays, continuum model
external injection locking, 82–92
without external injection, 68–81
planar arrays, continuum models
Cartesian coupling, external injection locking, 112–118
Cartesian coupling, without external injection, 104–109
triangular and hexagonal coupling schemes, 123–137
planar oscillator arrays, 51–54
L-band oscillators, receive-array experimentation, 203–207
Least-mean-square (LMS) algorithm, adaptive coupled-oscillator array beamforming, 319–320
Least-squares optimization, beamforming, coupled oscillator arrays, 300–301
Lindstedt method, coupled oscillators, averaging theory, 232–235
Linear arrays
beamforming, coupled oscillator arrays, multi-beam pattern generation, 312–315
beam-steering via end detuning, 93–95
beam-steering via end injection, 95–102
external injection locking, 81–92
nearest-neighbor coupling, 68–81
coupled oscillators, discrete model solution, 31–37
experimental validation of, 177–188
Linear matrix inequality, beamforming, coupled oscillator arrays, 299–301
Linear programming, beamforming, coupled oscillator arrays, 300–301
Load resistance, coupling network parameters, 54–66
“Locking range,” injection locking phenomenon, 10–21
Lyapunov exponents
beamforming, coupled oscillator arrays, 300–301
nonlinear dynamical systems
perturbation models, 223
Mathematical biology, early work, 3–5
Matrix equations, perturbation models, steady state and stability analysis, 242–246
Metal semiconductor field-effect transistor (MESFET) VCOs, phase noise experimentation, 212–213
Microwave mixer circuits, coupled oscillator arrays, conversion matrix, 267–268
Middle element injection
coupled oscillator arrays, alternative
perturbation models, external
coupled oscillators, numerical analysis, five-element array, 286–288
Minimization of the maximum, beamforming, coupled oscillator arrays, 301
Mirror-image spectra, injection locking phenomenon, 15–21
Mode-locked oscillators
linear array experimentation, 178–188
unlocked state experiments, 213–215
Modulation architectures, perturbation models, coupled oscillator arrays, 256–257
Monolithic microwave integrated circuit (MMIC)
linear array experimentation, 177–188
planar-array experimentation, 191–201
Multi-beam pattern generation, coupled-oscillator arrays, 311–315
Multichannel phase comparator, seven-element linear arrays, 180–188
Multiple time scales, perturbation models, 230–231
Multiport passive coupling networks, basic parameters, 55–66
Mutual injection locking
Narrow-band coupling networks, parameters, 55–66
Natural logarithm function, injection locking phenomenon, 12–21
Nearest-neighbor coupling
central pattern generator, 3–5
coupling delay
continuum models, causality violation, 148–159
linear arrays
beam steering via end injection, 99–102
planar oscillator arrays, 50–54
Near-zone radiated fields, planar-array experimentation, 195–201
Negative admittance, van der Pol oscillator model, 6–7
Negative resistance
coupling network parameters, 60–66
single capacitor coupling network, 62
Negative susceptance, coupling networks, 61–66
Neimark-Sacker bifurcation, nonlinear dynamical system, periodic orbits, 229
Neumann boundary condition
coupling delay, beam-steering, continuum model, 161–172
linear arrays, continuum modeling
external injection locking, 84–92
without external injection, 71–81
planar arrays, continuum models
Cartesian coupling, external injection locking, 112–118
Cartesian coupling, without external injection, 104–109
triangular and hexagonal coupling schemes, 123–137
Newton-Raphson algorithm
coupled oscillator arrays
harmonic-balance simulation, 270–272
transient numerical simulation, 265–266
planar arrays, continuum models, Cartesian coupling, external injection locking, 114–118
Nine-element oscillator array, planar-array experimentation with, 188–195
Noise correlation matrix, perturbation models, phase noise, 252–256
Nonautonomous continuous time dynamic systems, perturbation models, 220–222
Non-Cartesian coupling topologies, planar arrays, continuum models, 118, 121–137
Nondispersive transmission line, coupling delay and, 139–141
Nonlinear dynamical systems
equilibrium point bifurcation, 226–228
periodic orbit bifurcation, 228–229
nonlinear systems, 223
periodic steady state, 224–225
stability analysis, 223
Nonlinear first-order differential equations, coupled oscillators, 29–31
steady-state phase distribution stability, 41–46
Nonlinear optimization, self-oscillating mixers, numerical simulation, 293–296
Normalization, complexity in, 96
Norton equivalent circuits, coupling network parameters, 57–66
Numerical simulation, coupled oscillator arrays (COAs), 263–296
continuation methods, 269
envelope transient simulation, 268–269
externally injection-locked five-element linear coupled oscillator array, 285–288
five-element linear coupled oscillator array, 278–285
harmonic balance simulation, 266–267
periodic steady-state solutions, 270–272
harmonic radiation, extended scanning range, 288–291
periodic steady-state solutions, 270–272
self-oscillating mixer, 291–296
voltage-controlled oscillator, 272–278
Off-diagonal elements, single capacitor coupling network, 62
One-dimensional coupling network, unit cell examples, 60–62
One-dimensional root finding, planar arrays, continuum models, Cartesian coupling, external injection locking, 114–118
One-port equivalent circuits, coupled oscillator arrays
alternative perturbation models, 238–242
harmonic-balance simulation, 270–272
Open circuit voltage, coupling network parameters, 57–66
Optimization methods
beamforming, coupled oscillator arrays, 297–301
coupled oscillator arrays, stability analysis, 309–311
Orthogonality
coupled oscillators, discrete model solution, 35–37
planar oscillator arrays, 52–54
Oscillators
history of, 3
injection locking phenomenon, 8–21
mutual injection locking, two oscillators, 21–26
Over-steering, coupled oscillators, discrete model solution, 37
Partial differential equations
coupling delay, continuum models, causality violation, 149–159
planar arrays, continuum models, triangular and hexagonal coupling schemes, 125–137
Pattern nulling optimization, coupled oscillator systems, beamforming, 306–307
Pattern-synthesis convex optimization, beamforming, coupled oscillator arrays, multi-beam pattern generation, 311–315
Peak gain dynamics, planar arrays, continuum models, Cartesian coupling, without external injection, 106–111
Period-doubling bifurcation, nonlinear dynamical system, periodic orbits, 229
Periodic orbit
onlinear dynamical system bifurcations, 228–229
Poincare mapping, 222
Perturbation models
averaging method and multiple time scales, 230–231
coupled oscillator systems
externally injection-locked COAs, 247–250
coupled phase-locked loops, 258–261
dynamical systems
nonlinear systems
equilibrium point bifurcation, 226–228
periodic orbit bifurcation, 228–229
stability analysis, 223
nonlinear systems, 223
periodic steady state, 224–225
five-element arrays, harmonic balance simulation and, 284–285
matrix equations, steady state and stability analysis, 242–246
modulation, coupled-oscillator arrays, 256–257
van der Pol oscillator model, 235–237
Phased-array antenna elements, linear array experimentation, 179–188
Phase dynamics. See also Steady-state phase distribution
beamforming, coupled oscillator arrays, 301–307
coupled oscillator arrays, phase-locked loops, 258–261
coupling delay
beam-steering, continuum model, 166–171
five-element arrays, numerical simulation, 279–285
linear arrays, continuum modeling, beam steering via end injection, 95–102
linear arrays without external injection, 72–81
planar array experimentation, 188–201
planar arrays continuum models, Cartesian coupling, external injection locking, 111–118
self-oscillating mixers, receive-array experimentation, 207–210
Phase-locked loops (PLL), coupled oscillator arrays, perturbation models, 258–261
Phase measurement systems
planar-array experimentation, 191–201
seven-element linear arrays, 180–188
Phase noise
experimentation protocols and results, 210–213
Phase solution stability, coupled oscillators, full nonlinear formulation, 41–46
Phasor signal voltage, coupling delay, 140–141
Planar oscillator arrays
Cartesian coupling with external injection, 109–118
Cartesian coupling without external injection, 103–109
non-Cartesian coupling topologies, 118, 121–137
experimentation protocols, 188–201
Poincare mapping, perturbation models, dynamical systems, 222
Polarization agile, beam-scanning coupled oscillator array architecture, planar array experimentation, 196–201
Pole locations
coupling delay, beam-steering, continuum model, 162–172
linear arrays, continuum modeling
beam steering via end injection, 98–102
external injection locking, 86–92
Projected aperture loss
coupling delay, beam-steering, continuum model, 165–172
planar arrays, continuum models
Cartesian coupling, external injection locking, 118–120
Cartesian coupling, without external injection, 108–109
hexagonal coupling schemes, 130–137
triangular coupling schemes, 126–130
Propagators, coupling delay, discrete model, 144–147
Pseudomorphic high electron mobility transistor (pHEMT), numerical simulation, 272–278
Quadratic
coupled oscillators, steady-state phase distribution solution, 40–41
linear arrays, continuum modeling, external injection locking, 87–92
planar arrays, continuum models, triangular and hexagonal coupling schemes, 130–137
Quality factor (Q)
coupled oscillators, averaging theory, 233–235
coupling network parameters, 54–66
van der Pol oscillator model, perturbation parameters, 236–237
Quasi-optical techniques, linear array experimentation, 178–188
Quasi-periodic solutions, perturbation models, dynamical systems, 221–222
Radiation patterns, self-oscillating mixers, receive-array experimentation, 207–210
Radiofrequency signals, self-oscillating mixers, numerical simulation, 292–296
Receive-array experimentation, protocols and results, 201–210
Recurrence relations, coupled oscillators, discrete model solution, 33–37
Reference frequency, coupling delay, 139–141
Reflection coefficient, coupling delay, beam-steering, continuum model, 161–172
Residue calculus, coupling delay, beam-steering, continuum model, 162–172
Resistance, inductance, capacitance (RLC) resonator, averaging theory, coupled oscillators, 231–235
Round-trip transit time, coupling delay, continuum models, causality violation, 148–173
Saddle-node bifurcations, nonlinear dynamical systems, equilibrium points, 226–228
Scaled time, coupling delay, continuum models, 148–159
Scanned beam display, seven-element linear arrays, 182–188
Second-order differential equations, linear arrays without external injection, 68–81
Self-oscillating mixers (SOMs)
receive-array experimentation, 207–211
Sequential beam steering, planar arrays, continuum models, Cartesian coupling, external injection locking, 118, 121
Seven-element linear arrays, linear array experimentation, 180–186
Short circuit current, coupling network parameters, 57–66
Sidelobe reduction, receive-array experimentation, 203–206
Sine function, coupled oscillators, 28–31
Single branch-point solution, coupling delay, continuum model, 158–160
Single capacitor coupling network, parameters of, 62–66
Spatial power combining, linear array experimentation, 178–188
Spectral components, injection locking phenomenon, 15–21
Stability analysis
coupled-oscillator steady-state solution, 308–311
dynamical systems, perturbation models, 220–222
nonlinear dynamical systems, perturbation models, 220–226
periodic steady state, 224–225
Steady-state ensemble frequency. See Ensemble frequency
Steady-state phase distribution
continuation simulations, 269
full nonlinear formulation, solution stability, 41–46
coupling delay, beam-steering, continuum model, 166–172
linear arrays, continuum modeling
beam steering via end detuning, 93–95
beam steering via end injection, 98–102
external injection locking, 82–92
without external injection, 75–81
planar arrays, continuum models
Cartesian coupling, without external injection, 106–109
triangular and hexagonal coupling schemes, 124–137
Steady state solutions
coupled oscillator arrays
adaptive coupled-oscillator array beamforming, 317–320
harmonic balance simulation, 270–272
perturbation models
coupled oscillator arrays, 238–242
nonlinear dynamical systems, 223–226
Strong coupling
five-element arrays, numerical simulation, 280–285
phase noise experimentation, 211–213
Sturm-Liouville problems, linear arrays without external injection, 81
Substrate-integrated waveguide (SIW) technology, linear array experimentation, 184–189
Sum of residues, planar arrays, continuum models, Cartesian couplling, external injection locking, 115–118
Symmetric detuning, planar arrays, continuum models, Cartesian coupling, without external injection, 107–109
Synchronization
historical background, 3
self-oscillating mixers, numerical simulation, 293–296
voltage-controlled oscillators, numerical simulation, 273–278
Taylor series
coupling delay
beam-steering, continuum model, 163–172
linear arrays, continuum model
external injection locking, 81–92
without external injection, 68–81
planar arrays, continuum models
Cartesian coupling, without external injection, 103–109
non-Cartesian topologies, 123–137
Temporal discontinuities, coupling delay, continuum models, causality violation, 152–159
Three-by-three oscillator array, experimentation protocols, 188–195
Three-dimensional representation, coupling delay, beam-steering, continuum model, 166–172
Time-dependent perturbation
coupled oscillators, steady-state phase distribution stability, 42–46
Time-domain algorithms, coupled oscillator arrays, transient numerical simulation, 266
Transcendental equation
coupled oscillators, external injection locking, 47–50
linear arrays, continuum modeling, external injection locking, 86–92
Transmission line theory
linear array experimentation, 177–188
Triangular coupling topologies, planar arrays, continuum models, 118, 121–137
Tridiagonal admittance matrix
coupling network parameters, 55–66
single capacitor coupling network, 62
linear arrays
without external injection, 72–81
planar arrays, continuum models, Cartesian coupling, without external injection, 107–109
Turning-point bifurcation, nonlinear dynamical systems, equilibrium points, 226–228
Two-dimensional planar arrays
Cartesian coupling
with external injection, 109–118
without external injection, 103–109
experimentation using, 188–201
Unidirectional coupling, phase noise experimentation, 212–213
Uniform-strength injection signals, planar arrays, continuum models, Cartesian couplling, external injection locking, 116–118
Unlocked state, coupled oscillator experimentation, 213–215
van der Pol oscillator model
averaging method and multiple time scales, 230–231
historical development of, 5–7
injection locking phenomenon, 9–21
Very high frequency (VHF) transistor Colpitts oscillators, linear array experimentation, 177
Virtual instrument computer display, seven-element linear arrays, 181–184
Voltage-controlled oscillators
tuning bias, phase noise experimentation, 210–213
Voltage ratios, single capacitor coupling network, 62
Weak coupling approximation
coupled oscillators, averaging theory, 233–235
coupling network parameters, 58–66
single capacitor coupling network, 62
Zero-delay limit, coupling delay, branch point locations, 158–159